Wave energy converters (WECs) are devices that extract energy from waves in an open body of water (e.g., the ocean). Due to the immensity of the oceans and the continuous movements of the waves therein, the amount of energy corresponding to waves is substantial. However, the wave energy resource is spatially, temporally, and energetically variable. For example, energy within a particular region of a body of water is dependent upon a predominant frequency of waves, wave height, and width of the wave frequency spectrum.
Several algorithms have been developed in the literature that search for the optimal solution to the control of WECs in order to maximize the energy conversion. The frequency domain analysis of a WEC heaving buoy system leads to the criterion for maximum energy conversion—known as the Complex Conjugate Control (C3) that provides a means to compute the optimal float velocity, regardless of the spectral distribution of the excitation force. See J. Falnes, Ocean Waves and Oscillating Systems—Linear Interactions Including Wave-Energy Extraction, Cambridge University Press, NY, USA, (2002). This C3, however, is not causal which means a prediction for the wave elevation or the excitation force is needed for real time implementation. One implementation uses a feedforward control assuming the availability of an excitation force (wave) model. See S. Naito and S. Nakamura, “Wave energy absorption in irregular waves by feedforward control system,” Hydrodynamics of Ocean Wave-Energy Utilization, International Union of Theoretical and Applied Mechanics, Springer, Berlin, Heidelberg, pp. 269-280 (1986). Another feedback implementation computes the control force using both measurements and wave prediction data. See U. A. Korde et al., “Time domain control of a single mode wave energy device,” Proceedings of the Eleventh International Offshore and Polar Engineering Conference, Stavanger, Norway, pp. 555-560 (2001); and U. Korde, Ocean Eng. 26(7), 625 (1999). A velocity-tracking approach can also be used to implement the C3 where estimates of the excitation force are used to compute the optimal float velocity (through the feedforward loop) which is imposed on the WEC through a feedback loop. See C. Maisondieu and A. Clement, “A realizable force feedback-feedforward control loop for a piston wave absorber,” 8th International Workshop on Water Waves and Floating Bodies, St John's, Newfoundland, Canada, pp. 79-82 (1993). In all these C3 implementations, a prediction for the wave elevation and/or the wave excitation force is necessary.
Constraints on motions and forces, however, have motivated researchers to look for solutions in the time domain. In general, the solution of the constraint optimization problem is different from that of the unconstrained C3. The basic latching and declutching control strategies are attractive in that they do not require reactive power. See J. Hals et al., J. Offshore Mech. Arctic Eng. 133(1), 011401 (2011). In latching, the optimum oscillation phase is achieved by holding the absorber fixed during parts of the cycle. In clutching, it is achieved via coupling and decoupling the machinery at intervals. See G. Nolan et al., “Optimal damping profiles for a heaving buoy wave energy converter,” Proceedings of the Fifteenth International Offshore and Polar Engineering Conference ISOPE, Seoul, Korea (2005); and J. Hals et al., J. Offshore Mech. Arctic Eng. 133(3), 031101 (2011). Babarit et al. show that clutching is theoretically better than pseudocontinuous control that has a linear damping effect. See A. Babarit et al., Ocean Eng. 36(12-13), 1015 (2009). Clement and Babarit investigate the use of discrete control over continuous control, for latching control, declutching control, and the combination of both. The latter gives better results than each one individually; and the discrete control is always better when it is absolute, switching instantaneously from one model to the other. See A. H. Clement and A. Babarit, Philos. Trans. R. Soc. Lond. A: Math. Phys. Eng. Sci. 370, 288 (2012). Allison et al. apply a direct transcription approach to maximize the energy extraction. The results show that the direct transcription method generates a latching behavior for the cases with power constraints, while the declutching behavior only results when the tether goes slack. See J. T. Allison et al., “Wave energy extraction maximization using direct transcription,” ASME 2012 International Mechanical Engineering Congress and Exposition, Design, Materials and Manufacturing, Parts A, B, and C, vol. 3. ASME, Houston, Tex., USA, pp. 485-495 (2012).
Hals et al. compare various control strategies for a point absorber, including velocity-proportional control, approximate C3, approximate optimal velocity tracking, and model predictive control. See J. Hals et al., J. Offshore Mech. Arctic Eng. 133(1), 011401 (2011). The Model Predictive Control (MPC) methods use a discrete time model for predicting the states in the future to form the objective function for energy optimization. Kracht et al. compare several control strategies experimentally, including Proportional-Integral (PI) control and MPC. See P. Kracht et al., “First results from wave tank testing of different control strategies for a point absorber wave energy converter,” 2014 Ninth International Conference on Ecological Vehicles and Renewable Energies (EVER), Monte-Carlo, Monaco, pp. 1-8 (2014). The authors have found that MPC can significantly improve energy absorption when compared to the PI control. However, MPC needs a reliable estimation of the incoming incident wave, and the performance improvement is sensitive to the quality of the wave estimation. A PID control is used by Beirao et al. in which the controller gains are optimized for certain wave environments using information about the excitation force. See P. Beirao et al., “Identification and phase and amplitude control of the archimedes wave swing using a PID and IMC,” Second International Conference on Electrical Engineering, Coimbra, ISEC (2007). A variety of feedback control laws were developed using the C3 optimality conditions in Nielsen et al. See S. R. Nielsen et al., Ocean Eng. 72, 176 (2013). For example, the optimal velocity trajectory can be estimated, via wave estimation, and used along with the actual velocity in a feedback control system that aims at tracking the estimated optimal velocity. A linear quadratic Gaussian optimal control can be used to track optimal velocity as in Scruggs et al. See J. Scruggs et al., Appl. Ocean Res. 42, 1 (2013). One of the relatively recent WEC control optimization methods that can accommodate constraints on the control and the states is dynamic programming. See G. Li et al., Renew. Energy 48, 392 (2012). A prediction for the wave is needed when using dynamic programming, and a discretization for the time and space domains makes the computational cost of the method feasible for real time implementation. Another time domain strategy that can also handle constraints on both the control and the states is the pseudo-spectral method. In pseudo-spectral methods, the system states and control are assumed as series of basis functions, and the search for the solution is conducted using the assumed approximate functions. See G. Bacelli et al., “A control system for a self-reacting point absorber wave energy converter subject to constraints,” Proceedings of the 18th IFAC World Congress, Milano, Italy, pp. 11387-11392 (2011). A shape-based approach was recently developed for WEC control where a series expansion is used to approximate only the buoy velocity; this method can also accommodate motion constraints. See O. Abdelkhalik et al., “Control optimization of wave energy converters using a shape-based approach,” ASME Power & Energy 2015, ASME, San Diego, Calif. (2015); and O. Abdelkhalik et al., J. Ocean Eng. Mar. Energy 2(4), 473 (2016). A key optimality criterion is to make the buoy oscillation in phase with the excitation force. Fusco and Ringwood present a time domain control that meets this criterion and maintains the amplitude of the oscillation within given constraints. See F. Fusco and J. V. Ringwood, IEEE Trans. Sustain. Energy 4(1), 21 (2013). A nonstationary harmonic approximation for the wave excitation force is used. The controller tunes the ratio between the excitation force and the velocity in real-time for performance and constraints handling. A performance close to C3 and to MPC is achieved. Recently, an adaptive wave-by-wave control was developed such that the oscillation velocity closely matches the hydrodynamically optimum velocity for best power absorption. Such control requires prediction of the wave profile using up-wave measurements. See U. Korde et al., “Approaching maximum power conversion with exergy-based adaptive wave-by-wave control of a wave energy converter,” Proceedings of MTS/IEEE OCEANS, Genova, Italy (2015). In a more recent feedforward implementation, Korde investigates wave-by-wave control of a WEC using deterministic incident wave prediction based on up-wave surface measurement. See U. Korde, Appl. Ocean Res. 53, 31 (2015).
However, a need remains for a WEC that can efficiently extract wave energy over a full range of wave frequencies and that does not require prediction of the wave excitation force.